I often wonder what happens to the side-characters who are found and then dumped in their future. Do they connect with their descendants? If so, how does that turn out? What sort of jobs do they take? How do they go about their day-to-day lives?
There have been a few stories where we see this idea in the present, where someone in the past (or in a coma) comes to consciousness in our time period. This concept also is used to show dystopian sci-fi societies.
I think it would be interesting to explore the life of a "normal" person who somehow wakes up in a "normal" future. What would that person do? How would others react?
The story changes quite a bit depending on how far this time-traveler goes. There are plenty of stories that could come from this seed.